Presenting a solution with a stroke of genius

Question

When presenting the $3$-dimensional proof of the Desargues' theorem an average student might have, speaking informally, a "WTF moment".

It is an extreme case, but a similar situation could happen in an ordinary setting: what is interesting for gifted students might be inconceivable for the rest of the class. In fact, it might be a very good student presenting his solution.

• How to present a solution which contains an element which could not have been conceived by an average student?
• Is it possible to avoid negative afterthoughts?

Edit: Some more examples for clarification.

The Cantor's diagonal argument seems conceivable (thanks to hindsight bias): all he did was to construct a number which was different from all the others.

The definition of continuous function also seems conceivable if the student understands what we are trying to achieve.

A solution to a geometry problem might seem conceivable if the sequence of transformations was "reasonable" in each step, and no unknown element was added (if there was a line added, then it connected two, already existing points, perhaps also having some relation to some other element which is already there).

On the other hand, if a geometry problem requires adding several lines/points with no apparent reason behind them and shines only after all of them have been drawn, such solution is far to alien and no line of thought could convince the students that it could be, one day, in their reach.

Answer 1

I think it helps to make it abundantly clear whether or not you would expect an average student in your course to come up with such a "stroke of genius". If you're presenting something that might induce such a "WTF?!" moment, be very up front about it. I even like to put myself on the students' level if it's a particularly ingenious stroke, and will say something like, "I think this is beautiful, and even I would never ever come up with this, but that doesn't keep me from appreciating its ingenuity!"

Whenever possible, I also try to give some indication of why the person might have thought to do what they did. Maybe I can motivate/explain a few of the steps but not the bigger picture, or the entire sequence. For example, I've shown classes some of Euler's proofs of the Basel Problem, and those are monstrosities of ingenuity, but they have followable ideas within them. For instance, the mathematical legality of factoring sine as an infinite product sails over their heads, but they understand roots and linear factors. Likewise, the changes of variables Euler introduces are insanely clever, but the students can appreciate taking the log of both sides, because that converts the infinite product into an infinite sum, and we're trying to prove something about a series ...

So, ultimately, context and expectations are key. Do you expect the students to follow the idea but not come up with it? Say so. Do you expect them to follow the idea and get a glimmer of why one would think to do it? Say so. Do you expect them to be able to come up with this on their own? Say so. All along, marvel with them at the ingenuity of others.

Answer 2

Solutions that could not have been concieved by the average student

It is important to realise that many more solutions can't be concieved by the average student than we usually realise. Even solutions we think are quite routine can seem like strokes of genius to students new to the area.

Students often say to me, "I can follow the lecturer's proof/solution, but I have no idea how I would come up with something like that myself." And they really can't. In first year it's integrals and proofs about subspaces they struggle with. In second year it's "basic" proofs about limits using the $\epsilon$-$\delta$ definition. The problem is that our insights are based on a fairly deep understanding of the area and experience with lots of these proofs which has given us a familarity with the sorts of things that might help. The students do not have this experience, so they can't imagine even things that to us seem basic.

What the purpose of showing a solution is

I think before we answer the question, we should probably consider what the purpose of showing a solution is at all. Is it to convince the student of the truth of a statement? Is it to show them how to go about coming up with solutions? Is it to show them the patterns of argumentation that are acceptable in your discipline? Is it to gain an appreciation for the clever method of solution? Or some combination of these? Each of these purposes will give you a slightly different approach to presenting solutions, though they will overlap.

If it is students presenting the solutions, what is the purpose of that? Is it so that they get better at their communication skills? Are the students they present to supposed to have come to an understanding already, or are they supposed to use what the presenting student does to understand? Is it the correctness of the solution that is being assessed or the effectiveness of the presentation? Is the effectiveness judged by how the other students understand or on the structure of the solution itself? You need to consider these things so you can advise students on what sorts of things you want them to do when they present, so you can choose what sort of feedback you give them, and so you can decide when to intervene.

In most courses, the ultimate goal is usually to get students to solve problems on their own. Also, from my above observations about students, they need a lot of support to learn to do this. Thereforre, in my opinion the main reason for showing any solution is actually to help the students learn how to solve problems themselves.

Some suggestions

In light of this, I have some suggestions for things you can do.

Firstly, for all solutions, no matter how routine they seem, talk about the thought processes that help make the decision of what to do. Compare the given information to the goal, look up definitions of terms, make reference to other solutions that are similar, keep looking to the goal to see if you can do something to get closer.

Sometimes, you should try something that doesn't work, followed by some reasoning as to how you could tell in advance next time that it wouldn't work. You can also list some different things to try and make a quick assessment of them to choose the one that will work, rather than actually trying them out.

For students presenting, you can tell them the purpose is to describe how they came up with the solution, and you can make sure you ask them how they came up with the idea when they do present. You can also ask other students if they had a different solution. (I remember a tutorial when one of my students used two blackboards to do a proof, and when I asked the shocked audience if they had another way, another student came up and showed the intended three-line proof. It was a great lesson in different approaches for all of us.)

For things that really are strokes of genius, even for us who are experienced, there does come a point where you can say that people worked on this for quite some time and after a lot of false starts they came up with THIS idea, which is really a very brilliant inspiration you couldn't predict. There are still some morals for the students' own problem-solving though: they should persevere when problem-solving because you might have to make some false starts before hitting upon the right idea; more experience with solutions in a particular area make it more likely to come up with a helpful idea; going outwards to a bigger domain (like going up to 3D) can sometimes help.

The important thing for me is to be clear that you can learn about problem-solving from solutions to problems, even when it is a stroke of genius.

Answer 3

Yes, I think most arguments are "routine" in the sense of being very similar to others in the relevant context, while now-and-then there is something "out of the blue".

The latter can be deconstructed a bit, by admitting that all the goofy attempts that failed were not shown, but only the one that succeeded. Thus, possibly, if all the other attempts were depicted as well, the eventual success might seem not so disconnected from the "routine" line of thought, or, at least, might not seem so magical with a preamble of many failures. (And one should also point out that "failures" are usually informative, and in this sense mathematics is experimental, rather than "deductive", despite our collective mythology.)

Answer 4

A strange question. Almost every time you show something new to the students, it will be something that would never have occurred to them. Try to remember the first time you saw the derivation of the formula for the quadratic proved, or the $\epsilon - \delta$ definition of limit, or Cantor's diagonal argument. So this isn't the exception, it is the norm in the classroom. Part of what makes it rewarding, and also what can be so frustrating.

Answer 5

In the case of Desargues' theorem, and many others, the solution is not so unexpected when you know the history. Desargues was one of the inventors of projective geometry, whose original goal was to develop a mathematical theory of perspective drawing. He also worked as an architect, where he prepared accurate perspective drawings from blueprints. If you've ever taken an art class where they teach you how to do perspective drawing, they'll teach you to take each family of parallel lines in your space and choose some vanishing point somewhere on your easel to draw them passing through. If you have many families of parallel lines in coplanar directions, than these "points at infinity" will lie on a "line at infinity". Check out this YouTube drawing instruction video for the sort of art lesson I am talking about.

For example, here is Escher's Inside St. Peters (1935). See how all the vertical lines pass through one point down below the edge of the paper, all the north-south floor tiles converge at a second point and all of the east-west floor tiles converge at a third. (I just realized this wasn't the best example to choose: These three points at infinity aren't colinear because the three families of parallel lines don't lie in coplanar directions. See around 8:40 in the above linked video for a better example..)

It is not so strange that some one who was intimately familiar with translating between figures in three space and geometry of lines on the page would look at a figure and see it as a sketch of a three dimensional figure. Specifically, seeing $(\overline{AB}, \overline{ab})$, $(\overline{AC}, \overline{ac})$ and $(\overline{BC}, \overline{bc})$ as pairs of parallel lines in coplanar directions, and the line on which $\overline{AB} \cap \overline{ab}$, $\overline{AC} \cap \overline{ac}$, $\overline{BC} \cap \overline{bc}$ lie as the line at infinity.

In this particular case, I wonder if more is true: I wonder if Desargues invented the problem in order to show off how powerful projective geometry is! One of the easiest ways to solve a problem is to start with the solution.

You can't always take the time to pull back the curtain and talk about motivation like this, but I've found that students really appreciate it when you can.